On Error Estimate of an Approximate Method to Solve an Inverse Problem for a Semi-Linear Differential Equation

Abstract

An inverse problem for a semi-linear differential-operator equation in a Hilbert space
is considered in the paper. The projection regularization method is used to get a stable
approximate solution to the nonlinear ill-posed problem. The regularization parameter is
chosen referring to the Lavrentev scheme. A sharp error estimate of the considered method
on a correctness class defined by means of a nonlinear operator is obtained. The value of
the continuity module for the corresponding problem on the correctness classes plays an
important role in the investigation of the methods for the solution of ill-posed problems
in order to state their optimality. The linear operators are used, as a rule, to define the
correctness classes. The two-sided estimate of the continuity module for the nonlinear inverse
problem on the correctness class defined by a nonlinear operator is obtained in the present
work. The obtained estimate of the continuity module is used to prove the order-optimality
of the projection regularization method on the analyzed correctness class.